How is the measurement applied in daily life?

The glass and its secrets

Glasses are part of everyday life. What we naturally use in our everyday lives still puzzles physicists. How are glasses made? Or how do they behave at low temperatures? Siegfried Hunklinger from the Kirchhoff Institute for Physics reports on the ingenious methods that scientists have to use to coax some of their best-kept secrets out of the glasses.

Glasses have been used in daily life for a long time. However, many of their physical properties are still largely not understood. This includes in particular the process of their creation, the so-called glass transition, and the behavior of glasses at low temperatures - the topic of this article.

What does "low temperatures" mean? As is well known in physics, temperature is a measure of the strength of atomic motion. At absolute zero (T = 0 K) all atoms are at rest in the classic sense. Expressed on the usual temperature scale, this corresponds to a temperature of -273.15 ° C. Thermodynamics teaches us that one can only approach absolute zero experimentally, but never reach it. Which temperatures actually occur in nature? The illustration on page 11 gives an overview: The logarithmic scale representation starts with the temperature of the hottest stars at 109 K and ends at 10-6 K, the lowest temperature reached in the laboratory. The small point in the middle of the scale on page 11 ("Organic Life") reflects the extremely narrow temperature range in which our life takes place.

The following considerations are limited to processes that take place below 1 K. Strangely enough, such low temperatures are not found in nature, because the lowest temperature in the universe is 2.7 K. It is determined by the background radiation that comes from the Big Bang. Without the targeted use of special cooling mechanisms, this temperature cannot be fallen below anywhere. Physical phenomena that we only observe below 2.7 K in the laboratory therefore do not occur in nature unless intelligent beings are able to generate locally lower temperatures.

In the past, solid-state physics was primarily concerned with crystals, which are characterized by the regular arrangement of their atoms or molecules. A look at the illustration on page 11 (top left) shows that there is only one, well-defined arrangement of atoms for every perfect crystal.

The high symmetry of the crystals greatly facilitates the mathematical description of their properties. In the case of glass-like solids, however, the situation is quite different. Since glasses are usually made by rapidly cooling their melts, the solidification takes place so quickly that the atoms do not have enough time to arrange themselves regularly. Small cavities of atomic size are formed. Therefore every glass - even with the same composition - will have a different atomic structure.

It is true that the majority of the atoms in glass also have a clear equilibrium position in which they, like the atoms of crystals, perform thermal oscillations. But there are always configurations in which the position of individual atoms or entire groups of atoms is not clearly defined. This is illustrated by the top and right figures on this page. They are almost equivalent and only differ in the displacement of two atoms marked in blue. If you look at the two atoms on their way from one equilibrium position to the other, they move in an energy or potential landscape, as shown in the figure on page 12 (middle). In fact, at higher temperatures the atom constantly changes its position of rest due to the thermal movement: It jumps over the mountain of potential of the double well potential from one equilibrium position to another. If classical physics were unreservedly valid, the atom would eventually come to rest in one of the two hollows when it cools down. However, quantum mechanics allows - albeit with a low probability - to "tunnel through" the mountain of potential. As a result, atomic movements over relatively large distances are possible in glasses even at very low temperatures. Crystals, on the other hand, solidify completely at low temperatures, as each atom is firmly in its rest position.

Schematic two-dimensional representation of the structure of a crystal (left) and of glass (middle, right). Highlighted in dark blue is an atom that can assume two different positions of equilibrium.

The logarithmic temperature scale shows some characteristic phenomena.

The actual conditions are somewhat more complex: In reality, not only individual atoms move, but also small groups with a few atoms. One speaks therefore of "tunnel systems" in which an unspecified "particle" moves, the exact atomic structure of which is unknown. Since the trough depths generally differ slightly due to the irregular structure, there is an energy difference between the two states even without taking tunneling into account. The tunnel movement increases this difference, or causes a difference if the two troughs are the same depth. Regardless of the special circumstances, the particle can only have two well-defined energy values, which is why one speaks of "two-level systems". Since the energy landscape and the size of the tunneling particle in the irregular glass structure differ from system to system, their energy splitting also varies. Near absolute zero, the two-level systems are in an energetically lower state. At finite temperatures, the tunnel systems absorb thermal energy: They are excited.

The measurement of the specific heat provides a first indication of the existence of tunnel systems in glass and their absence in perfect crystals. It indicates which energy must be supplied to a sample in order to bring about a certain increase in temperature. Its value is a measure of the number of vibrations or structural rearrangements that can be thermally occupied. The figure on the right on page 12 shows the temperature profile of this measured variable for crystalline quartz and for quartz glass (on a logarithmic scale). The difference between the two curves becomes more and more pronounced with decreasing temperature, until at 25 mK the specific heat of the glass is about 5000 times greater than that of the crystal. This result shows that in glasses, even at low temperatures, other degrees of freedom can be excited in addition to the atomic vibrations.

Specific heat of quartz glass and crystalline quartz as a function of temperature.

In this context it should also be mentioned that this behavior is not a peculiarity of quartz glass, but that such a high specific heat is found in all glasses. Surprisingly, not only the temperature profile, but also the absolute value of the specific heat is similar in all glasses, regardless of the chemical composition: high-purity quartz glass, window glass, metallic glasses and vitreous polymers contain roughly a tunnel system for every million atoms.

Double well potential with energy levels.

We demonstrated that two-level systems exist almost three decades ago with the help of ultrasound measurements at frequencies around 1 GHz and temperatures below 1 K. This was achieved by observing the saturation behavior of the ultrasonic absorption. A clear proof that the two-level systems of the glasses cannot be described classically and are actually caused by quantum mechanical tunnel movements can be provided by the generation of "coherent echoes". For this purpose, the sample is first cooled to temperatures below 50 mK in order to reduce the consequences of the undesired thermal movement. Then a microwave is radiated into the glass sample for a short time. Their alternating electrical field stimulates the charged tunnel particles to oscillate in time with the wave. An oscillating electrical polarization is associated with the common oscillation, which can be detected using suitable measurement methods.

The influence of a weak magnetic field on the dielectric constant of a multi-component glass

If you switch the microwave off again, the tunnel systems continue to oscillate, but get out of step within a few nanoseconds. This means that the measurable macroscopic polarization of the glass sample disappears. After a few microseconds, the microwave is switched on a second time. If you choose the strength and duration of the two microwave pulses in a suitable way, you will find that after some time an electrical polarization builds up spontaneously, which can be measured and disappears again after a short time. It is found that the time between the second pulse and the occurrence of the "echo" is exactly the same as between the first and second pulse.

A closer look shows that this phenomenon - which, by the way, is well known from experiments on nuclear magnetic resonance - can only occur if the phase of all resonating tunnel systems has a well-defined value at all times (i.e. also between the two microwave pulses). From this it must be concluded that the phase of the quantum mechanical wave function of the tunnel systems also develops in a predictable way during the entire observation period. This clearly shows that the two-level systems are caused by quantum mechanical tunneling processes. With increasing temperature, the tunnel systems are more and more disturbed by the increased thermal movement of the environment, so that hardly any more echoes can be generated. The tunneling process becomes less important until the movement of the particles in the double trough potentials follows the laws of classical physics.

In the second part of the article I would like to address new investigations in which the dielectric constant of glasses was measured. As a reminder: the dielectric constant is a measure of the strength of the electrical polarization that is caused in the sample by an electrical field. Since tunnel systems can carry an electrical charge, they not only largely determine the elastic and thermal properties of glass at low temperatures, but also the behavior of the dielectric constant. But what influence should a magnetic field have on the dielectric constant?

Three-dimensional potential ("Mexican hat") with two hollows

If one consults classical theories, one expects at best a very weak quadratic decrease in the dielectric constant with the applied field. This prediction has been confirmed by previous experiments: At field strengths of one Tesla, which correspond to about 20,000 times the earth's field, relative changes of at most 10-5 were reported. In our new experiments, the dielectric constant of a glass consisting of BaO, Al2O3 and SiO2 was investigated. The figure on page 12 below shows the result of a measurement at 64 mK. Surprisingly, the dielectric constant initially rises sharply with the magnetic field, passes through a pronounced maximum, which is followed by a second, weaker one. Measurements at different temperatures show that this effect increases with decreasing temperature.

Variation of the dielectric constant of a multi-component glass as a function of the magnetic field

First of all, it is astonishing that the glass responds to magnetic fields at all, as it does not contain any magnetic components. Even if the observations are not fully understood, approaches exist to explain this phenomenon. It is assumed that the tunneling particle does not move in a straight line, but moves along a curved path. The potential could then (as shown in the illustration on page 12/13) have the shape of a Mexican hat with two hollows on the brim. The tunneling particle then has two ways to get from one potential well to the other.

When a magnetic field is applied, the so-called Lorentz force acts on the charged particle, which influences the tunnel movement. Quantum mechanics teaches us that both paths are no longer equal in the magnetic field and that interference effects that lead to a change in the energy splitting of the two-level systems must be taken into account. Since the energy splitting also influences the dielectric constant, this consideration makes it understandable that the magnetic field can actually change this measured variable. With the help of the interference effects, the occurrence of the oscillations can in principle also be understood ("Aharanov-Bohm effect").

Although this explains the observations in principle, the enormous size of the effect and the fact that relatively small magnetic fields already give rise to oscillations remain incomprehensible. If one uses typical numerical values ​​in the theory, one finds that oscillations should only occur in magnetic fields that are a hundred thousand times larger than those actually used. Another effect must therefore play an important role, namely the coupling of the movement of the tunnel systems to one another.

The occurrence of such coupling is understandable when one considers that the tunneling particles are charged. The tunnel systems thus carry an electrical dipole moment. If the coupling through the resulting electric fields is sufficiently strong, the tunnel movement of the individual particles is "synchronized". The common movement increases the contribution of the individual tunnel systems and the oscillation of the dielectric constant takes place in the case of much smaller fields. The picture developed here also explains the observation that the magnetic field dependence of the dielectric constant disappears at higher temperatures and becomes more and more apparent with decreasing temperature. It is obvious that the interaction between the tunnel systems can only be of importance if the thermal energy is comparable to or less than the coupling energy. According to our knowledge, this condition is only met below 0.1 K.

What do you expect when the sample cools down further and further? Even if there are no systematic studies on this yet, there are already some remarkable results. A kink in the temperature profile of the dielectric constant was observed at 5.8 mK. This suggests that the dynamic behavior of the tunnel systems changes significantly at this temperature.

The results of a measurement in a magnetic field at a temperature of only 1.85 mK are shown in the figure below on page 13. This experiment was carried out in a so-called nuclear spin demagnetization cryostat, in which temperatures well below 1 mK are reached. The sample was in a place where the magnetic field was initially only about 20 µT (microtesla), that is, it was weaker than the earth's magnetic field. As the upper half of the figure shows, the field was slowly changing. Although the changes corresponded to only about a fifth of the earth's field, the dielectric constant of the glass follows this field variation in a measurable way. Obviously, the effect discussed above increases with decreasing temperature to such an extent that even the smallest magnetic fields have an influence on the dynamics of the tunnel systems.

It is possible that at these low temperatures a noticeable fraction of the existing tunnel systems execute a correlated movement, that is, the tunneling particles of these systems all move at the same rate. Such a joint movement of a large number of atoms has not yet been observed in any solid.

While the low-temperature properties of perfect dielectric crystals are well understood, glasses always hold surprises in store. The reason for this are the tunnel systems, the existence of which is a result of the irregular structure of the glasses. Although a fundamental understanding of these phenomena has been developed over the past two decades, a number of questions still remain unanswered. This includes the question of the microscopic nature of the tunnel systems. Furthermore, it is not understood why the chemical composition of the glasses has no influence on the number of existing tunnel systems.

Completely new perspectives are opened up by experiments in which the influence of magnetic fields is investigated. Phenomena occur that have not been observed or expected in glasses or other solids before. The coupling to magnetic fields can be so strong that even fields weaker than the earth's field have significant effects on the dynamic properties of the glass being examined.

Prof. Dr. Siegfried Hunklinger
Kirchhoff Institute for Physics, Albert-Ueberle-Strasse 3-5, 69120 Heidelberg,
Telephone (0 62 21) 54 92 61, fax (0 62 21) 54 92 62